21 PRODUCT OF MEASURES 225
(13.21.6). Hence from (13.9.13), it follows that for each xf N = N, u N
the mapping y ^ h(x, y) is /i-integrable, and therefore the function
/»
is defined almost everywhere with respect to L The fact that it is A-inteerahl^
and the relation (13.21.7.1), then followfrom (13.21.4.2), and^^(1321.4.3)
of (13.21.7), ' ™' the
(13.21.7.2) (fh(X,y)dl
But here again it needs to be said that the right-hand sides of (13 21 71
(13.21.7.2) can be defined and equal without h being v-integrable (even
v-measurable) (Problem 3). "'egraoie (even
(13.21.8) Leth^Obea v-measurable function. Then the mapping
XH+ h(x,y)dn(y)
is ^.-measurable, and
(13.21.8.1)
= J
Let (KJ i be an increasing sequence of compact subsets of X x Y which
cover X x Y (3.18.3). Then we have h = sup hn, where h. = inf(/,;
, , '; on the other hand (13.5.7)
= sup v(/0. By (13.6.2) and (13.21.7), there exists a A-negligible set N
such that, for all x$ N, all the functions y^hn(x,y) are ^-integrable; also
(13.21.7) the functions x^fax.y) d^(y) are A-integrable and we have
v(AJ = jWx)jh,,(x, y) driy)'for all «. Hence it follows from (13.5.7) that
j h(x, y) dn(y~) = sup | hn(x,
for all X t N. Consequently the function x»j*h(X, y) dtfy) is A-measurable
by (13 9.11), and the relation (13.21.8.1) follows by another application of
(TJ.D./).be positive measures on X